The answer to this question is positive. Indeed, one can use Cartan's criterion for solvability [1]} which states that a matrix Lie algebra \(\mathfrak {g}\)  is solvable (in characteristic zero) if and only if \(K(\mathfrak {g}, [\mathfrak {g},\mathfrak {g}]) =0\) , where \(K\)  designates the Killing form\(K(u,v)=\operatorname{Tr}({\rm ad}(u){\rm ad}(v))\) . of \(\mathfrak {g}\) . This is also equivalent to \(\operatorname{Tr}(uv)=0\)  for all \(u \in \mathfrak {g}\)  and all \(v \in [\mathfrak {g},\mathfrak {g}]\) , the derived Lie algebra of \(\mathfrak {g}\) .
